The linear equation `3x-5y=15` has

Options

A. a unique solution
B. two solutions
C. infinitely many solutions
D. no solution

Explanation

To determine the nature of the solutions of the linear equation \(3x - 5y = 15\), we can analyze it step by step.

### Step 1: Identify the Equation
The given linear equation is:
\[ 3x - 5y = 15 \]

### Step 2: Rearranging the Equation
We can rearrange the equation to express \(y\) in terms of \(x\):
\[ -5y = -3x + 15 \]
\[ y = \frac{3}{5}x - 3 \]

### Step 3: Identify the Slope and Y-Intercept
From the equation \(y = \frac{3}{5}x - 3\), we can identify:
- The slope \(m = \frac{3}{5}\)
- The y-intercept \(b = -3\)

### Step 4: Finding Points on the Line
To understand the solutions, we can find two points on the line by substituting values for \(x\) and solving for \(y\).

1. **When \(x = 0\)**:
\[ y = \frac{3}{5}(0) - 3 = -3 \]
So, the point is \((0, -3)\).

2. **When \(y = 0\)**:
\[ 0 = \frac{3}{5}x - 3 \]
\[ \frac{3}{5}x = 3 \]
\[ x = 5 \]
So, the point is \((5, 0)\).

### Step 5: Graphing the Equation
By plotting the points \((0, -3)\) and \((5, 0)\), we can draw the line representing the equation \(3x - 5y = 15\).

### Step 6: Analyzing the Solutions
A linear equation in two variables represents a straight line in the coordinate plane. Since this line extends infinitely in both directions, it means:

- For every value of \(x\), there is a corresponding value of \(y\).
- Therefore, there are infinitely many solutions to the equation.

### Conclusion
The linear equation \(3x - 5y = 15\) has infinitely many solutions.

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